VGT is probably doable with all classes at least (fighter and mybe warlord being the only unknowns for me, and I'm wondering about Half-Dragons because of Golems). It inovolves scumming for certain gods on the first floor, while prepping a different god topside for the trickier propositions.
Not sure about badges, though. I've yet to manage the vicious token run - the strongest approach to VGT (Gorgon. so far), seems to have a difficult enough early game for the difficulty to ramp up faster than you can get rolling in VVGT. I'm probably way out of shape though, and if I had a few days to sink into it, maybe I'd discover something.
As far as other badges go -
Faithless is doable as long as the vampire is doable.
Parched I think I got with the Rogue.
Purist is probably a matter of scumming up the right god combo (although it also means no Whoopaz, Dragonshield or prep dependent classes (Rogue's one of those), and no altar topside). Not sure if doable, but it's free to scum, so I'm sure someone will pull it off eventualy once we figure out what exactly's the strongest class in there.
Warmonger - I've herd of people doing old Gaan'Telet warmonger (and Naga City), and it always came as a surprise to me. Getindare, and several other glyphs, seem too integral to my playstyle. It has it's benefits in free inventory slots that I never have otherwise, possible early burst piety with GG, Drac or Taurog, CP and so on...
Hoarder/Miser - it's probably doable with a resist stacker of some sort who can get all he needs from glyphs, gods and preps. Miser definitely easier than hoarder, hoarder is pure downside, with miser you can at least sink gold into TT or a prepped/stolen CB or Alch scroll (small consolation though). A hoarder Wizard of some sort might be the way to go.
As far as veto slots go - I vetoed a big bunch of spelcasting stuff for the run, mostly quest items. Can't acess the game right now, but if anyone's interested I can look into it and post my reasoning.
EDIT: As for vicious token vgt run candidates so far: Rogue, Paladin, Wizard, and Monk seem to be what pwned regular VGT the hardest. Many others haven't been tried, but as far as I know Gorgon has an easy but weird time in there, as does the Vamp but the difficulty spike from the token probably kills them. Transmuter and Priest have been done, but it's unlikely they could do a VT run.
I'm having trouble assessing actual difficulty because I haven't actually been at it for a while, and while I was, I killed myself on the firaball retaliation floor due to innatention so often that I don't have a proper grasp of how strong certain things are...
If no "auto-broken" shannenigans are just flat out broken enough to let you breeze through it, for example if Drac Paladin with VGT bloodpool levels isn't simply strong enough to beat it with little real effort or if you can't build a max res monk fast enough, so it's a matter of careful play and trickshots - I think the Wizard might actually be the guy with the biggest advantage in being able to use more glyphs, leverage CP from all of them, have more inventory space, have a cheaper GETINDARE and glyph scouting to save exploration and let him plan better. Plus, you can go almost any route with him as far as gods are concerned. Sidestepper might have insight on this, I'd love to hear about his prep checklist for a VVGT wizard run (or a regular one for that matter).